Terminal Velocity of a Falling Object

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The definition of terminal velocity is, according to, “The velocity at which a falling body moves through a medium, as air, when the force of resistance of the medium is equal in magnitude and opposite in direction to the force of gravity.”

The terminal velocity of a falling object can be determined using Newton’s Second Law of Motion, which is the force exerted on an object equals the product of the mass of an object times acceleration of the object (F=MxA). Solving for the acceleration A will put the equation in the needed form: A=F/M. Since the object is falling through the air, the force F is the difference between the weight of the object W and air resistance D (W-D). Substituting W-D for F obtains the equation A=(W-D)/M.

The formula for air resistance uses the drag coefficient CD, atmospheric density ρ, velocity in the air of the object V, and reference area α. The formula is D=CDxρxV^2xα/2. If the weight is small, the air resistance will quickly be equal to the weight of the object. Since W=D, W-D=0. Substituting this in the equation obtains A=0/M=0, so the acceleration is 0. An object with zero acceleration will have constant velocity. This constant vertical velocity is called terminal velocity.

Since D=W, substituting W for D in the equation for air resistance and solving for V will obtain the terminal velocity. The original equation is D=CDxρxV^2xα/2, substituting W for D obtains W=CDxρxV^2xα/2. Multiplying by two results in 2xD=CDxρxV^2xα. Dividing by everything in the formula except V^2 results in (2xD)/(CDxρxα)=V^2.

Taking the square root of both sides of the formula obtains V=[(2xD)/(CDxρxα)]^1/2, or the terminal velocity is equal to the square root of twice the air resistance D divided by the drag coefficient CD times the atmospheric density ρ times the reference area α.  

The drag coefficient CD is determined using an equation  based on an object falling in the air. Solving for CD in the formula obtains CD=D/ρxαxV^2/2, or in words the drag coefficient is equal to the atmospheric density ρ times reference area α times the terminal velocity squared divided by two.

The reference area α depends on what is being calculated. If the terminal velocity of an airplane is being calculated, the total surface area, frontal area, or wing area might be used. The amount of air resistance depends on the shape of the object and other factors. If the inclination is small, the air resistance is nearly constant. As the inclination increases, the air resistance increases. A small inclination is less than or equal to 5 degrees.  

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